Question #134788
Check whether x^5+9x^4+12x^2+6 is reducible over Q.
1
Expert's answer
2020-09-24T15:52:58-0400

Solution: We can check whether x5+9x4+12x2+6x^5+9x^4+12x^2+6 is reducible over Q or not by Eisenstein's criterion.

Let p=3p=3 (a prime number). We know that ana_n is the coefficient of highest power. Now, we notice that 3an=13 \nmid a_n=1, but since 39,312,363|9, 3|12, 3|6 and 303|0, i.e., 33 divides all the other coefficients of given polynomial.

We also notice that 32=96=a03^2=9 \nmid 6=a_0

Thus, Eisenstein's criterion is satisfied and so, x5+9x4+12x2+6x^5+9x^4+12x^2+6 is irreducible over Q[x].

Answer: x5+9x4+12x2+6x^5+9x^4+12x^2+6 is irreducible over Q[x].


Need a fast expert's response?

Submit order

and get a quick answer at the best price

for any assignment or question with DETAILED EXPLANATIONS!

Comments

No comments. Be the first!
LATEST TUTORIALS
APPROVED BY CLIENTS